Absolute stability conditions for some scalar nonlinear time-delay ...

Absolute stability conditions for some scalar
nonlinear time-delay systems with monotone
increasing nonlinearity ⋆
Daniela Danciu ∗
∗ Department of Automatic Control, University of Craiova, A.I.Cuza, 13,
Craiova, RO-200585, Romania (e-mail: daniela@automation.ucv.ro).
Abstract: This paper deals with the analysis of the absolute stability for a class of scalar nonlinear timedelay
systems having monotone increasing nonlinearities. The approach is based on frequency domain
inequalities of Popov type for time-delay systems in the critical case of the transfer function with a simple
zero pole. In order to solve the minmax problem which arises from the frequency domain inequality, the
analytical analysis was completed by numerical computations using MATLAB software package. We
have obtained a time-delay dependent absolute stability condition.
Keywords: Absolute stability, Frequency domain inequality, Limit stability, Time-delay system
1. STATE OF THE ART AND PROBLEM STATEMENT
1.1 Absolute stability and frequency domain inequalities
The above remark will be useful in the next sections. In the sequel
we shall use the following notations regarding the signals
in Fig. 1: µ := µ 1 = −σ 2 and σ := σ 1 = µ 2 .
The property of absolute stability had been motivated by Letov
(1955); Letov and Lurie (1957) by the rather poor information
about nonlinearity and, as Răsvan (2002) remarks, “in a more
contemporary statement this is nothing else but robust stability
with respect to some kind of nonlinear function uncertainty”.
Absolute stability refers to the global asymptotic stability of the
zero equilibrium of the nonlinear system
ẋ(t) = Ax − bϕ(c ∗ x) (1)
having sector restricted nonlinearities of the form (see Fig. 2)
0 ≤ ϕ ≤ ϕ(σ)
σ
≤ ϕ ≤ +∞ , ϕ(0) = 0 (2)
the property of the equilibrium being valid for all the linear and
nonlinear functions verifying (1).
For the absolute stability problem, the system under the analysis
(1) might be written as a “negative” feedback connection (Fig.
1) of the linear block L described by
Fig. 1. Absolute stability feedback structure.
ẋ(t) = Ax + bµ 1
σ 1 = c ∗ x
with the nonlinear block, subject to (2) and described by
(3)
σ 2 = ϕ(µ 2 ) (4)
the interconnection rules being
µ 2 = σ 1 , µ 1 = −σ 2 . (5)
⋆ This work has been partly supported by the Research Project CNCSIS ID-95
Fig. 2. Sector restricted nonlinearities
Concerning the system with the structure in Fig. 1 we recall
here the so-called Aizerman problem (Răsvan et al. (2010)).
Let L be a linear controlled system. If ϕ(σ) is a linear function
ϕ(σ) = hσ, any stability criterion would give a sector

h ∈ (ϕ H
,ϕ H ) called the Hurwitz sector which, corresponding to
the necessary and sufficiently stability conditions, is thus maximal.
If, on the other hand, one considers nonlinear functions
verifying (2), then only sufficient global stability conditions
can be obtained (generally speaking) and the resulting maximal
sector will be, as a rule, more narrow than the Hurwitz sector.
The comparison of the two sectors is the Aizerman problem:
the closer they are, the less is the “degree of conservatism”
obtained via the available sufficient conditions of absolute stability
(among which the frequency domain inequalities together
with the Liapunov functions and the Linear Matrix Inequalities
associated to them are the less conservative).
According to Lefschetz (1965), the development of the absolute
stability theory has known two periods: the pre-Popov period
and Popov period that started after 1960. Regarding the second
period, this is marked by the frequency domain inequalities
used for analyzing the stability of nonlinear systems. These
were introduced by the Romanian scientist V. M. Popov in his
pioneering paper (Popov (1959)) and became known worldwide
especially after his seminal paper (Popov (1961)).
For the goal of the paper we shall introduce here the absolute
stability result based on frequency domain inequality of Popovtype
for time-delay systems in the critical case of a zero root
(a straightforward extension of the Theorem 6.1 in Răsvan
(1975)).
Theorem 1. Consider the control system described by
ẋ(t) = Ax(t) +
σ(t) = c ′ 0x(t) +
r
∑
1
r
∑
1
B k x(t − τ k ) + b 0 ξ (t) +
˙ξ (t) = −ϕ(σ(t)),
c ′ k x(t − τ k) + γ 0 ξ (t) +
r
∑
1
r
∑
1
b k ξ (t − τ k ),
γ k ξ (t − τ k )
where ϕ(σ) is a continuous nonlinearity verifying the conditions:
0 < ϕ(σ)
σ
(6)
< k ≤ +∞ , ϕ(0) = 0. (7)
One supposes that the characteristic equation
det
(
sI − A −
has all its roots within C − , and
γ 0 +
r
∑
1
γ i −
(
c ′ 0 +
r
∑c ′ j
1
)(
r
∑
1
A +
B k e −sτ k
r
∑
1
If there exists q ≥ 0 finite such that
where
)
B k
) −1 (b 0 +
= 0 (8)
r
∑
1
b l
)
> 0.
(9)
1
+ Re (1 + ıωq)H(ıω) ≥ 0, (10)
k
H(s) = 1 s
(
[
sI − A −
γ 0 +
r
∑
1
r
∑
1
B k e −sτ k
γ i e −sτ i
+
) −1 (
(
c ′ 0 +
b 0 +
r
∑
1
r
∑
1
c ′ je −sτ j
)
·
b l e −sτ l
) ⎤ ⎦,
(11)
then the system (6) is asymptotic stable for every nonlinearity
within the class above defined.
Remark 2. The Theorem 1 ensures absolute stability for all
nonlinearities which verify (7), yielding the absolute stability
sector (0,k) which can be compared with the Hurwitz sector in
the Aizerman problem; the fulfilment of the frequency domain
inequality (10) will give the “dimension” of the absolute stability
sector. The inequality (9) ensures the limit stability property
of the linear block in the critical case considered and, is similarly
to the condition (12) in Theorem 4 from the subsection
Limit stability property which will follow. The equation (11) is
nothing else that the transfer function of the system (6).
1.2 The limit stability property
In connection with absolute stability and Aizerman problem
is the limit stability property - introduced by Aizerman and
Gantmakher (1963). Consider again the systems having the
structure in Fig. 1 and suppose that the nonlinear function is
known with some uncertainty. As already said, we call absolute
stability a robust version of the stability property, i.e. the
stability of the zero equilibrium for all nonlinear and linear
function within the sector (0,k) and belonging to a certain
class of functions. For ensuring the absolute stability property, a
necessary and minimal condition would be exponential stability
for a single linear function of the sector. In particular, if L
defines an exponentially stable linear system, the property
(called by Popov (1973) minimal stability) holds for ϕ(σ) ≡ 0.
But if L is in a critical case i.e. it has the spectrum in C − as
well as on ıR a necessary (and minimal) requirement would be
exponential stability for ϕ(σ) = εσ with 0 < ε < ε 0 and ε 0 > 0
arbitrarily small. This is called limit stability and is in fact a
property of the linear block L; the rigorous definition would be
as follows
Definition 3. Let L be a linear dynamical block described by
some input/output operator - a convolution (in the time domain)
or a transfer function (in the complex domain) - connecting
the input µ to the output σ. System L is said to have the
limit stability property if it is output stabilizable by the output
feedback µ = −εσ with 0 < ε < ε 0 and ε 0 > 0 arbitrarily small.
There are known necessary and sufficient conditions for limit
stability for linear block L associated to a strictly proper rational
transfer function - see Aizerman and Gantmakher (1963). They
were extended to “the delay case”, i.e. to linear blocks whose
transfer functions are strictly proper meromorphic functions
defined by a ratio of quasi-polynomials; for the sake of the
completeness, we shall give in the sequel the main result for
this case (Răsvan (1975)).
Theorem 4. Consider a linear block with the transfer function
H(s) = N(s)/D(s) where N(s) and D(s) are quasi-polynomials,
H(s) being strictly proper in the sense that the degree of the
principal term of D(s) is larger than that of N(s) and assume
D(s) to have at most a finite number of roots on ıR, the other
roots being in C − . For the limit stability of this system it
is necessary and sufficient that the multiplicity of each pole
should be at most 2 and the following conditions hold for it

a) for a simple zero pole
Res H(s)| s=0 > 0 (12)
b) for a simple non-zero pole, if the Laurent expansion is
considered
Re H(ıω) =
e −1
+ d 0 − e 1 (ω − ω 0 )−
ω − ω 0
−d 2 (ω − ω 0 ) 2 + ...
Im H(ıω) = −d −1
ω − ω 0
+ e 0 + d 1 (ω − ω 0 )−
−e 2 (ω − ω 0 ) 2 + ...
(13)
and the sequence: d −1 ; e −1 e 0 ; −d 1 ; −e 1 e 2 ; d 3 ; ... is associated,
either d −1 ≠ 0 or e −1 ≠ 0 and the first non-zero term of the
above sequence should be strictly positive;
c) for a double zero root, if the Laurent expansion is considered
Re H(ıω) = −d −2
ω 2 + d 0 − d 2 ω 2 + ...
Im H(ıω) = −d −1
ω + d 1ω − d 3 ω 2 + ...
(14)
and the sequence −d −1 ; d 1 ; −d 3 ; ... is associated, then i)
d −2 > 0 and ii) the first non-zero term of the sequence should
be strictly negative;
d) for a double non-zero root, if the Laurent expansion is
considered
−d −2
Re H(ıω) =
(ω − ω 0 ) 2 − e −1
+ d 0 −
ω − ω 0
−e 1 (ω − ω 0 ) − ...
−e −2
Im H(ıω) =
(ω − ω 0 ) 2 − d −1
+ e 0 +
ω − ω 0
+d 1 (ω − ω 0 ) − ...
(15)
and the sequence d −1 ; −e 2 0 ; −d 1; −e 2 2 ; ... is associated, then
i) d −2 > 0, e −2 = 0 and ii) the first non-zero element of the
sequence should be strictly positive.
Within the theorem the notion “principal term” is used in the
sense of Pontryagin: if
h(z,w) = ∑ a mn z m w n
m,n
is a polynomial in two variables, a rs z r w s is the principal term of
the polynomial if a rs ≠ 0 and for any other term with a mn ≠ 0
one of the following is possible: i) r > m, s > n; ii) r = m, s > n;
iii) r > m, s = n. Note that any quasi-polynomial may be given
the form h(z,e z ) with h(z,w) a polynomial in two variables.
1.3 The mathematical model and the problem statement
where ψ(·) is a monotone increasing nonlinearity, τ > 0 is a
constant time-delay and the initial condition x(θ) = ϕ(θ), for
θ ∈ [−τ,0], where ϕ ∈ C (−τ,0;R).
The differential equation (16) may model, for instance, the fluid
dynamics in high-performance networks. We refer here to the
mathematical model proposed in Kelly (2001) for describing
the rate control algorithm in order to avoid the congestion in
communication networks. Considering the case when a collection
of flows uses a single resource and shares the same gain
parameter K, the model of Kelly (2001) reads as
ẏ(t) = K[w − y(t − τ)p(y(t − τ))] , K > 0 , w > 0 (17)
where τ > 0, assumed constant, represents the round-trip time
and p(·) can be viewed as “the probability a packet produces a
congestion indication signal”, thus being “positive, continuous,
strictly increasing function of y and bounded above by unity”
(Kelly (2001)).
We shall turn now to the mathematical model (16). Let x be
the unique equilibrium of (16), thus verifying a = bψ(x)x.
Using a change of the coordinates, ξ = x − x, one can shift the
equilibrium point x to the origin so that the system (16) can be
written into the form:
where the nonlinear function
verifies the conditions
˙ξ = −bφ(ξ (t − τ)) (18)
φ(ξ ) = ψ(x + ξ )(x + ξ ) − ψ(x)x. (19)
φ(0) = 0 , φ(ξ )
ξ
> 0. (20)
Instead (18), we shall consider - without loss of the generality -
the nonlinear system
˙ξ = −φ(ξ (t − τ)) (21)
with τ > 0 and ξ (θ) = χ(θ), for θ ∈ [−τ,0], where χ ∈
C (−τ,0;R).
The aim of this paper is to obtain conditions for absolute
stability of the class of systems described by (21), where the
nonlinearity φ(·) is a monotone increasing function verifying
(20).
2. THE ABSOLUTE STABILITY RESULT
2.1 The frequency domain condition
Following the absolute stability approach which is described in
section I, the system (21) can be written as a negative feedback
connection of the linear block
Consider the class of scalar nonlinear time-delay systems described
by
ẋ(t) = a − bx(t − τ)ψ(x(t − τ)) , a > 0 , b > 0 (16)
˙ξ (t) = µ(t)
σ(t) = ξ (t − τ)
with a nonlinear one, the interconnection rule being
(22)

µ(t) = −φ(σ(t)). (23)
On the other hand, the system (21) is of the form (6) with
A = b 0 = c 0 = γ 0 = 0 and B k = b k = c k = 0 for k = 1,r,
γ 1 := γ = 1. In both cases the transfer function of the linear
block results
H(s) = e−τs
, (24)
s
and one remarks the system is in the critical case a) of the
Theorem 4: a simple zero pole. It can be easily seen that the
linear block has the limit stability property since
Res H(s)| s=0 = 1 > 0 (25)
or, equivalent, it is verified the condition (9).
The frequency domain inequality (10) is written in our case as
1
+ Re (1 + ıωθ)e−ıωτ
k ıω = 1 (
k + θ cosωτ − sinωτ )
> 0
ω
(26)
and, dividing by τ > 0 one obtains
1
+ θ 0 cosλ − sinλ > 0 , ∀λ > 0 (27)
k 0 λ
where we denoted λ := ωτ, θ 0 := θ τ and k 0 := kτ. The inequality
(27) can be written as a minimax problem
Let
(
max min θ 0 cosλ − sinλ )
> − 1 . (28)
θ 0 ≥0 λ≥0
λ k 0
f (λ) = θ 0 cosλ − sinλ
λ
, ∀λ > 0 (29)
be the function under evaluation. On can see that f (0) = θ 0 −
1 and the condition f (0) > −
k 1 0
will give θ 0 > 1 −
k 1 0
. For
λ d → ∞ and λ d = nπ the most unfavorable case is cosλ d = −1
and f (λ d ) = −θ 0 > −
k 1 0
gives θ 0 <
k 1 0
. We have obtained the
general boundary conditions:
1 − 1 k 0
< θ 0 < 1 k 0
(30)
and one can see that the alternate sign of the function impose
as necessary the condition k 0 < ∞. Also, one remarks the
inequality (30) shows that the larger θ 0 is the smaller is k 0 and
as a consequence the narrow is the absolute stability sector.
A summary analysis shows that the interval (0,π) is interesting
for the variation of function f . First of all, one observes that the
second term in (29) is the function sinc(λ) = sinλ/λ whose
principal lobe has positive values on (0,π/2) and due to the
minus sign it has a unfavorable influence for our minimax
problem. On the other hand, for λ ∈ (π/2,π) both terms of the
function have negative values. Evaluating f (π/2) = −2/π >
−1/k 0 , ∀θ 0 we obtain an estimation for k 0 :
1 − 1 k 0
< 1 − 2 π < 2 π < − 1 k 0
. (31)
2.2 The analysis of the function on intervals
Concerning the minimax problem, the general analysis of the
function and its derivative
f ′ λ cosλ − sinλ
(λ) = −θ 0 sinλ −
λ 2 (32)
leads to the following remarks:
• λ = nπ: f (nπ) = (−1) n θ 0 which shows again that θ 0
cannot be too large since for n = (2p + 1), p = 0,1,...
it would give a more restrictive condition for the absolute
stability sector: k 0 <
θ 1 0
.
• λ = n
2 π : f ((2p + 1)π + 2 π ) > 0 and it is not important for
the problem; f (2pπ +
2 π ) = − 1
2pπ+ 2
π < 0 and the smallest
value (having the maximum modulus) is f (
2 π ) = − 2 π .
A. Consider now the interval λ ∈ (0,π).
a) lim λ→0+ f ′ (λ) = 0 which means λ = 0 is an extremum.
Making use of Taylor expansion we obtain f ′ (λ) = (
3 1 −θ 0)λ +
o(λ 2 ) and we deduce that on (0,π): i) if 0 ≤ θ 0 < 1 3 then λ = 0
is a minimum and ii) if θ 0 > 1 3
then λ = 0 is a maximum.
b) Evaluating the derivative in λ =
2 π we obtain an other point
4
of interest for our analysis on θ 0 : . For θ
π 2 0 = 4 , f ( π π 2 2 ) = − 2 π
is a minimum on (0,π) and f is strictly increasing for θ 0 < 4
π 2
and decreasing otherwise.
We thus have obtained the following intervals of interest for θ 0
when λ ∈ (0,π):
• θ 0 ∈ (0, 1 3 ): f (0) = θ 0 − 1 < 2 3
is a minimum on (0,π).
• θ 0 ∈ (
3 1, 4 ): f (0) is a maximum on (0,π) and there exists
π 2
a minimum in (0,
2 π ), then the function rises; f (π) =
−θ 0 < 0; f ′ (π) = 1 π > 0, ∀θ 0.
• θ 0 ∈ ( 4 ,∞): the minimum on (0,π) is within the interval
π 2
(
2 π ,π) and f (λ 0) < − 2 π .
B. The analysis on the intervals (2pπ,(2p + 1)π), p = 1,2,...
gives the following conclusions:
1
• θ 0 <
(2pπ+ 2 π < 1 )2 3
: f has a negative minimum within the
interval (2pπ,2pπ +
2 π ).
1
• θ 0 >
(2pπ+ 2 π : f has a negative minimum within the
)2
interval (2pπ +
2 π ,(2p + 1)π).
C. On the intervals ((2p + 1)π,(2p + 2)π), p = 1,2,... the
extremum points are positive maxima and, are not of interest
for our analysis.
We conclude that the intervals of interest for checking the
minima have the general form
λ ∈ (2pπ,(2p + 1)π), p = 0,1,.... (33)
Let λ p be the solution of f ′ (λ) = 0 on such an interval; it
verifies also

λ p
tanλ p =
1 − θ 0 λp
2
and one can compute the minimum on the interval
f (λ p ) = [ θ 0 (1 − θ 0 λ 2 p) ] sinλ
λ
(34)
< 0 (35)
Since on such intervals sinc(λ) > 0, it results that for p ≥ 1 this
minimum is negative no matter where it is placed: either within
the first or second half of the interval.
We can estimate now that the larger θ 0 > 0 is the smaller
the negative minimum will be. This means that θ 0 cannot be
increased too much since the minimax problem requires the
maximization of the minima with respect to θ 0 .
2.3 The analysis of the local minima on intervals
The sequence of the local minima is defined by the equation
(1 − θ 0 λ 2 )sinλ − λ cosλ = 0 (36)
with the solution defined by (34) and the value of a local
minimum
f (λ p ) =
(
θ 0 +
)
1
θ 0 λp 2 cosλ p . (37)
− 1
The analysis on each interval of interest for θ 0 , namely those
we have found in section 2.2 (A and B), is simple but tediously.
It reveals that the function of the minima is the same in all cases
and has the general form
g(x) = − 1 + θ 0(θ 0 x − 1)
√
(θ0 x − 1) 2 + x
where x = λ 2 p ≥ 0. The derivative of the “minima” function
gives the extremum point at
which means θ 0 ≥ 1 3 .
One has thus to solve
g ′ (x) = − (3θ 0 − 1) − θ 2 0 x
2[(θ 0 x − 1) 2 + x] 3 2
x = (3θ 0 − 1)
θ 2 0
⎧
⎫
⎨
max
θ 0 ≥ 3
1 ⎩ − 1 + θ 0(θ 0 λ0 2 √
− 1) ⎬
(θ 0 λ0 2 − 1)2 + λ0
2 ⎭
(38)
(39)
≥ 0, (40)
(41)
where λ 0 is the zero of (36) for a choice of θ 0 . One can use for
this purpose the mathematical software packages.
The significant local minima of the function f , computed by
using MATLAB software for θ 0 ≥ 1 3
are given in Table 1.
Fig. 3 and Fig. 4 (the zoom in on (0,π)) show the graphical
representations of f (λ) for both cases θ 0 ∈ [ 1 3 , 4 ] and θ
π 2 0 > 4 .
π 2
One can observe that, as we previously have estimated, the
values of minima decrease when θ 0 increases. On the other
hand, it can be seen that for any of these functions the smallest
minimum is within the interval (0,π). In order to solve (41) we
have thus to find the maximum of these absolute minima on
(0,π). The curve of the absolute minima of f (λ) with respect
to θ 0 is illustrated in Fig. 5; also, the significant values are in
Table I. We conclude that for the case θ 0 ∈ ( 1 3
,∞), the minimum
we search for is f min = −0.64968 obtained for θ 0 = 0.38.
Fig. 6 shows the curves of f (λ) for θ 0 ∈ [0, 1 3
) (the negative
value of θ 0 is taken from curiosity). These confirm that the
absolute minimum is in this case λ = 0 ∈ [0,π). But for this
case we have already obtained that f (0) = g(0) = θ 0 − 1 ∈
[−1,− 2 3 ), thus the maximum “absolute minimum” for θ 0 ∈
[0, 1 3
) is less than −0.64968, the value of the minimum for
θ 0 ∈ ( 1 3 ,∞).
We conclude that the solution of the minimax problem (28) is
−0.64968 ≥ − 1 k 0
(42)
and thus we have determined the sector of absolute stability
The result of our analysis reads as follow
k < 1.5392 . (43)
τ
Consider the time-delay nonlinear system (21) with φ(·) a
continuous monotone increasing nonlinearity. The system is
asymptotic stable for all nonlinear (and linear) functions verifying
(20) and which are within the sector (0, 1.5392
τ
).
Remark 5. Obviously, the absolute stability sector (43) is narrowed
by increasing the time delay. Regarding the mathematical
model (16) for the congestion problem in communication
networks this time-delay dependent absolute stability condition
means that the class of the nonlinear functions ψ(·) of the type
(7) is diminished by increasing the round-trip time τ.
3. CONCLUSIONS
This paper deals with the analysis of the absolute stability for a
class of scalar nonlinear time-delay systems having monotone
increasing nonlinearities. These systems can be encountered,
for instance, as congestion problems in high-performance communication
networks.
We have used an approach based on frequency domain inequalities
of Popov type for time-delay systems in the critical case of
the transfer function with a simple zero pole. In order to solve
the minmax problem which arises from the frequency domain
inequality, the analytical analysis was completed by numerical
computations using MATLAB software package.
We have obtained a time-delay dependent absolute stability
condition i.e. the class of the nonlinear functions, and therefore
of the nonlinear systems under consideration, can be limited by
large time-delays (for instance, the round-trip time in the case
of the congestion problems in high-performance communication
networks).

f
f
0.8
0.6
0.4
f(λ) for θ ∈ (0.37,0.5)
θ = 0.36
θ = 0.375
θ = 0.39
θ = 0.42
θ = 0.45
θ = 0.5
θ = 0.6
0.4
0.2
0
θ ∈ (−0.2, 0.32)
0.2
−0.2
0
−0.4
−0.2
−0.6
−0.4
−0.6
−0.8
−1
θ = − 0.2
θ = 0.1
θ = 0.2
θ = 0.32
−0.8
0 2 4 6 8 10 12 14 16 18 20
λ
Fig. 3. Function f for θ ∈ ( 1 3 , 1 2 ).
−0.5
−0.55
−0.6
−0.65
−0.7
f(λ) for θ ∈ (0.37,0.5)
0.5 1 1.5 2 2.5
λ
θ = 0.36
θ = 0.375
θ = 0.39
θ = 0.42
θ = 0.45
θ = 0.5
θ = 0.6
Fig. 4. Zoom in: function f for λ ∈ (0,π) and θ ∈ ( 1 3 , 1 2 ).
minima
−0.63
−0.64
−0.65
−0.66
−0.67
−0.68
−0.69
−0.7
The curve of the absolute minima
−0.71
0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
θ
Fig. 5. The absolute minima of f (λ) for θ ∈ ( 1 3 , 1 2 ).
REFERENCES
Aizerman, M.A. and Gantmakher, F.R. (1963). Absolute stability
of regulator systems. USSR Academy Publ. House,
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−1.2
0 2 4 6 8 10 12 14 16
Fig. 6. Function f for θ ∈ (−0.2, 1 3 ,).
Table 1. The maximum minima of f (λ) for θ ∈
( 1 3 , 4
π 2 )
θ 0 f (λ 0 )
0.33 -0.6702
0.34 -0.6608
0.35 -0.6548
0.36 -0.6513
0.37 -0.6497
0.38 -0.64968
0.39 -0.6510
0.40 -0.6532
0.41 -0.6563
0.42 -0.6602
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